2017 年高考数学（上海卷）

exams_raw/普通高考/2017/2017上海.pdf · HS-MATH-1024-v2.1-solution-aware

21 题0 个小问/题组

一、填空题 · 组合数学

已知集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{4}\htmlData{tutor-start=14,tutor-end=16}{\}}$ ，集合 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{4}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{5}\htmlData{tutor-start=11,tutor-end=13}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{=}$ \_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 1 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数学竞赛/待细分

若排列数 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{6}}^{\htmlData{tutor-start=7,tutor-end=8}{m}} \htmlData{tutor-start=10,tutor-end=11}{=} \htmlData{tutor-start=12,tutor-end=13}{6} \htmlData{tutor-start=14,tutor-end=21}{\times }\htmlData{tutor-start=21,tutor-end=22}{5} \htmlData{tutor-start=23,tutor-end=30}{\times }\htmlData{tutor-start=30,tutor-end=31}{4}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{=}$ \_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 2 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 代数

不等式 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=9}{1}}{\htmlData{tutor-start=11,tutor-end=12}{x}} \htmlData{tutor-start=14,tutor-end=15}{>} \htmlData{tutor-start=16,tutor-end=17}{1}$ 的解集为\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 3 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数学竞赛/待细分

已知球的体积为 $\htmlData{tutor-start=0,tutor-end=1}{3}\htmlData{tutor-start=1,tutor-end=2}{6}\htmlData{tutor-start=2,tutor-end=5}{\pi}$ ，则该球主视图的面积等于\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 4 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数学竞赛/待细分

已知复数 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=3}{+} \frac{\htmlData{tutor-start=10,tutor-end=11}{3}}{\htmlData{tutor-start=13,tutor-end=14}{z}} \htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{0}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{=}$ \_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 5 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数学竞赛/待细分

设双曲线 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{9}} \htmlData{tutor-start=16,tutor-end=17}{-} \frac{\htmlData{tutor-start=24,tutor-end=25}{y}^{\htmlData{tutor-start=27,tutor-end=28}{2}}}{\htmlData{tutor-start=31,tutor-end=32}{b}^{\htmlData{tutor-start=34,tutor-end=35}{2}}} \htmlData{tutor-start=38,tutor-end=39}{=} \htmlData{tutor-start=40,tutor-end=41}{1}$ ( $\htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0}$ ) 的焦点为 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为该双曲线上的一点，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{F}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{5}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{F}_{\htmlData{tutor-start=5,tutor-end=6}{2}}\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{=}$ \_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 6 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 平面几何

如图，以长方体 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{A}_{\htmlData{tutor-start=8,tutor-end=9}{1}}\htmlData{tutor-start=10,tutor-end=11}{B}_{\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{C}_{\htmlData{tutor-start=18,tutor-end=19}{1}}\htmlData{tutor-start=20,tutor-end=21}{D}_{\htmlData{tutor-start=23,tutor-end=24}{1}}$ 的顶点 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 为坐标原点，过 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 的三条棱所在的直线为坐标轴，建立空间直角坐标系，若 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{D}\htmlData{tutor-start=17,tutor-end=18}{B}_{\htmlData{tutor-start=20,tutor-end=21}{1}}}$ 的坐标为 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{4}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{)}$ ，则 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{C}_{\htmlData{tutor-start=20,tutor-end=21}{1}}}$ 的坐标是\_\_\_\_\_\_.

**原卷图示 1**原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 7 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 代数

定义在 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=11}{\infty}\htmlData{tutor-start=11,tutor-end=12}{)}$ 上的函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}$ 的反函数为 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}^{\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{)}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \begin{cases} \htmlData{tutor-start=21,tutor-end=22}{3}^{\htmlData{tutor-start=24,tutor-end=25}{x}} \htmlData{tutor-start=27,tutor-end=28}{-} \htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{,} & \htmlData{tutor-start=34,tutor-end=35}{x} \htmlData{tutor-start=36,tutor-end=40}{\le }\htmlData{tutor-start=40,tutor-end=41}{0}\htmlData{tutor-start=41,tutor-end=42}{,} \\ \htmlData{tutor-start=46,tutor-end=47}{f}\htmlData{tutor-start=47,tutor-end=48}{(}\htmlData{tutor-start=48,tutor-end=49}{x}\htmlData{tutor-start=49,tutor-end=50}{)}\htmlData{tutor-start=50,tutor-end=51}{,} & \htmlData{tutor-start=54,tutor-end=55}{x} \htmlData{tutor-start=56,tutor-end=57}{>} \htmlData{tutor-start=58,tutor-end=59}{0}\htmlData{tutor-start=59,tutor-end=60}{,} \end{cases}$ 为奇函数，则 $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{2}$ 的解为\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 8 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 代数

已知四个函数：① $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{x}$ ，② $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{-}\frac{\htmlData{tutor-start=9,tutor-end=10}{1}}{\htmlData{tutor-start=12,tutor-end=13}{x}}$ ，③ $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}^{\htmlData{tutor-start=5,tutor-end=6}{3}}$ ，④ $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}^{\frac{\htmlData{tutor-start=11,tutor-end=12}{1}}{\htmlData{tutor-start=14,tutor-end=15}{2}}}$ . 从中任选 2 个，则事件“所选 2 个函数的图象有且仅有一个公共点”的概率为\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 9 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数论

已知数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 和 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ ，其中 $a_{n}=n^{2}, n \in \mathbf{N}^$ ， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的项是互不相等的正整数，若对于任意 $n \in \mathbf{N}^$ ， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的第 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 项都等于 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的第 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 项，则 $\frac{\lg\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{b}_{\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{b}_{\htmlData{tutor-start=19,tutor-end=20}{4}} \htmlData{tutor-start=22,tutor-end=23}{b}_{\htmlData{tutor-start=25,tutor-end=26}{9}} \htmlData{tutor-start=28,tutor-end=29}{b}_{\htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{6}}\htmlData{tutor-start=34,tutor-end=35}{)}}{\lg\htmlData{tutor-start=40,tutor-end=41}{(}\htmlData{tutor-start=41,tutor-end=42}{b}_{\htmlData{tutor-start=44,tutor-end=45}{1}} \htmlData{tutor-start=47,tutor-end=48}{b}_{\htmlData{tutor-start=50,tutor-end=51}{2}} \htmlData{tutor-start=53,tutor-end=54}{b}_{\htmlData{tutor-start=56,tutor-end=57}{3}} \htmlData{tutor-start=59,tutor-end=60}{b}_{\htmlData{tutor-start=62,tutor-end=63}{4}}\htmlData{tutor-start=64,tutor-end=65}{)}} \htmlData{tutor-start=67,tutor-end=68}{=}$ \_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 10 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}_{\htmlData{tutor-start=8,tutor-end=9}{1}}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=18}{\alpha}_{\htmlData{tutor-start=20,tutor-end=21}{2}} \htmlData{tutor-start=23,tutor-end=27}{\in }\mathbf{\htmlData{tutor-start=35,tutor-end=36}{R}}$ ，且 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{+}\sin\htmlData{tutor-start=15,tutor-end=21}{\alpha}_{\htmlData{tutor-start=23,tutor-end=24}{1}}} \htmlData{tutor-start=27,tutor-end=28}{+} \frac{\htmlData{tutor-start=35,tutor-end=36}{1}}{\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{+}\sin\htmlData{tutor-start=44,tutor-end=45}{(}\htmlData{tutor-start=45,tutor-end=46}{2}\htmlData{tutor-start=46,tutor-end=52}{\alpha}_{\htmlData{tutor-start=54,tutor-end=55}{2}}\htmlData{tutor-start=56,tutor-end=57}{)}} \htmlData{tutor-start=59,tutor-end=60}{=} \htmlData{tutor-start=61,tutor-end=62}{2}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=7}{\pi }\htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=15}{\alpha}_{\htmlData{tutor-start=17,tutor-end=18}{1}} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=28}{\alpha}_{\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{|}$ 的最小值等于\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 11 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、填空题 · 平面几何

如图，用 35 个单位正方形拼成一个矩形，点 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{P}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{P}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{P}_{\htmlData{tutor-start=24,tutor-end=25}{4}}$ 以及四个标记为“▲”的点在正方形的顶点处，设集合 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=9}{\{}\htmlData{tutor-start=9,tutor-end=10}{P}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{P}_{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=24}{P}_{\htmlData{tutor-start=26,tutor-end=27}{3}}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{P}_{\htmlData{tutor-start=33,tutor-end=34}{4}}\htmlData{tutor-start=35,tutor-end=37}{\}}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{P} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=12}{\Omega}$ ，过 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 作直线 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ ，使得不在 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ 上的“▲”的点分布在 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ 的两侧. 用 $\htmlData{tutor-start=0,tutor-end=1}{D}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{l}_{\htmlData{tutor-start=9,tutor-end=10}{P}}\htmlData{tutor-start=11,tutor-end=12}{)}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{D}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{l}_{\htmlData{tutor-start=9,tutor-end=10}{P}}\htmlData{tutor-start=11,tutor-end=12}{)}$ 分别表示 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ 一侧和另一侧的“▲”的点到 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ 的距离之和. 若过 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的直线 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{P}}$ 中有且只有一条满足 $\htmlData{tutor-start=0,tutor-end=1}{D}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{l}_{\htmlData{tutor-start=9,tutor-end=10}{P}}\htmlData{tutor-start=11,tutor-end=12}{)}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{D}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{l}_{\htmlData{tutor-start=22,tutor-end=23}{P}}\htmlData{tutor-start=24,tutor-end=25}{)}$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 中所有这样的 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为\_\_\_\_\_\_.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 12 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、选择题 · 数学竞赛/待细分

关于 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}$ 的二元一次方程组 $\begin{cases} \htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{5}\htmlData{tutor-start=17,tutor-end=18}{y}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{0}\htmlData{tutor-start=20,tutor-end=21}{,} \\ \htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{x}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{3}\htmlData{tutor-start=29,tutor-end=30}{y}\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{4}\htmlData{tutor-start=32,tutor-end=33}{,} \end{cases}$ 的系数行列式 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 为 ( )

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 13 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、选择题 · 代数

在数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 中， $a_{n} = \left(-\frac{1}{2}\right)^{n}, n \in \mathbf{N}^*$ ，则 $\lim_{\htmlData{tutor-start=6,tutor-end=7}{n} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=18}{\infty}} \htmlData{tutor-start=20,tutor-end=21}{a}_{\htmlData{tutor-start=23,tutor-end=24}{n}}$ ( )

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 14 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、选择题 · 代数

已知 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ 为实常数，数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项 $x_{n} = an^{2}+bn+c, n \in \mathbf{N}^$ ，则“存在 $k \in \mathbf{N}^$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{k}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{x}_{\htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{0}\htmlData{tutor-start=16,tutor-end=17}{0}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{k}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{3}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{0}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{k}}$ 成等差数列”的一个必要条件是 ( )

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 15 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、选择题 · 平面几何

在平面直角坐标系 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中，已知椭圆 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{:} \frac{\htmlData{tutor-start=13,tutor-end=14}{x}^{\htmlData{tutor-start=16,tutor-end=17}{2}}}{\htmlData{tutor-start=20,tutor-end=21}{3}\htmlData{tutor-start=21,tutor-end=22}{6}} \htmlData{tutor-start=24,tutor-end=25}{+} \frac{\htmlData{tutor-start=32,tutor-end=33}{y}^{\htmlData{tutor-start=35,tutor-end=36}{2}}}{\htmlData{tutor-start=39,tutor-end=40}{4}} \htmlData{tutor-start=42,tutor-end=43}{=} \htmlData{tutor-start=44,tutor-end=45}{1}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{:} \htmlData{tutor-start=7,tutor-end=8}{x}^{\htmlData{tutor-start=10,tutor-end=11}{2}} \htmlData{tutor-start=13,tutor-end=14}{+} \frac{\htmlData{tutor-start=21,tutor-end=22}{y}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}{\htmlData{tutor-start=28,tutor-end=29}{9}} \htmlData{tutor-start=31,tutor-end=32}{=} \htmlData{tutor-start=33,tutor-end=34}{1}$ . $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 上的动点， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 上的动点， $\htmlData{tutor-start=0,tutor-end=1}{w}$ 是 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{O}\htmlData{tutor-start=17,tutor-end=18}{P}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{O}\htmlData{tutor-start=43,tutor-end=44}{Q}}$ 的最大值. 记 $\htmlData{tutor-start=0,tutor-end=7}{\Omega }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=11}{\{}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{Q}\htmlData{tutor-start=16,tutor-end=17}{)} \htmlData{tutor-start=18,tutor-end=19}{|} \htmlData{tutor-start=20,tutor-end=21}{P} \text{ \htmlData{tutor-start=29,tutor-end=30}{在} } \htmlData{tutor-start=33,tutor-end=34}{C}_{\htmlData{tutor-start=36,tutor-end=37}{1}} \text{ \htmlData{tutor-start=46,tutor-end=47}{上}}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{Q} \text{ \htmlData{tutor-start=59,tutor-end=60}{在} } \htmlData{tutor-start=63,tutor-end=64}{C}_{\htmlData{tutor-start=66,tutor-end=67}{2}} \text{ \htmlData{tutor-start=76,tutor-end=77}{上}}\htmlData{tutor-start=78,tutor-end=79}{,} \text{ \htmlData{tutor-start=87,tutor-end=88}{且} } \overrightarrow{\htmlData{tutor-start=107,tutor-end=108}{O}\htmlData{tutor-start=108,tutor-end=109}{P}} \htmlData{tutor-start=111,tutor-end=117}{\cdot }\overrightarrow{\htmlData{tutor-start=133,tutor-end=134}{O}\htmlData{tutor-start=134,tutor-end=135}{Q}} \htmlData{tutor-start=137,tutor-end=138}{=} \htmlData{tutor-start=139,tutor-end=140}{w}\htmlData{tutor-start=140,tutor-end=142}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 中 ( )

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 16 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 平面几何

如图，直三棱柱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{C}_{\htmlData{tutor-start=17,tutor-end=18}{1}}$ 的底面为直角三角形，两直角边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 的长分别为 4 和 2，侧棱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}}$ 的长为 5. (1) 求三棱柱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{C}_{\htmlData{tutor-start=17,tutor-end=18}{1}}$ 的体积； (2) 设 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 中点，求直线 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{M}$ 与平面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 所成角的大小.

**原卷题面及图示 1**原卷第 1 页 · question_region_fallback · 需复核

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 17 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 平面几何

已知函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \cos^{\htmlData{tutor-start=13,tutor-end=14}{2}} \htmlData{tutor-start=16,tutor-end=17}{x} \htmlData{tutor-start=18,tutor-end=19}{-} \sin^{\htmlData{tutor-start=26,tutor-end=27}{2}} \htmlData{tutor-start=29,tutor-end=30}{x} \htmlData{tutor-start=31,tutor-end=32}{+} \frac{\htmlData{tutor-start=39,tutor-end=40}{1}}{\htmlData{tutor-start=42,tutor-end=43}{2}}\htmlData{tutor-start=44,tutor-end=45}{,} \htmlData{tutor-start=46,tutor-end=47}{x} \htmlData{tutor-start=48,tutor-end=52}{\in }\htmlData{tutor-start=52,tutor-end=53}{(}\htmlData{tutor-start=53,tutor-end=54}{0}\htmlData{tutor-start=54,tutor-end=55}{,} \htmlData{tutor-start=56,tutor-end=59}{\pi}\htmlData{tutor-start=59,tutor-end=60}{)}$ . (1) 求 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 的单调递增区间； (2) 设 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 为锐角三角形，角 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 所对边 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}\sqrt{\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{9}}$ ，角 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 所对边 $\htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{5}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{0}$ ，求 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的面积.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 18 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

解答题 · 数学竞赛/待细分

根据预测，某地第 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ( $n \in \mathbf{N}^*$ ) 个月共享单车的投放量和损失量分别为 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ (单位：辆)，其中 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \begin{cases} \htmlData{tutor-start=22,tutor-end=23}{5}\htmlData{tutor-start=23,tutor-end=24}{n}^{\htmlData{tutor-start=26,tutor-end=27}{4}} \htmlData{tutor-start=29,tutor-end=30}{+} \htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{5}\htmlData{tutor-start=33,tutor-end=34}{,} & \htmlData{tutor-start=37,tutor-end=38}{1} \htmlData{tutor-start=39,tutor-end=49}{\leqslant }\htmlData{tutor-start=49,tutor-end=50}{n} \htmlData{tutor-start=51,tutor-end=61}{\leqslant }\htmlData{tutor-start=61,tutor-end=62}{3}\htmlData{tutor-start=62,tutor-end=63}{,} \\ \htmlData{tutor-start=67,tutor-end=68}{-}\htmlData{tutor-start=68,tutor-end=69}{1}\htmlData{tutor-start=69,tutor-end=70}{0}\htmlData{tutor-start=70,tutor-end=71}{n} \htmlData{tutor-start=72,tutor-end=73}{+} \htmlData{tutor-start=74,tutor-end=75}{4}\htmlData{tutor-start=75,tutor-end=76}{7}\htmlData{tutor-start=76,tutor-end=77}{0}\htmlData{tutor-start=77,tutor-end=78}{,} & \htmlData{tutor-start=81,tutor-end=82}{n} \htmlData{tutor-start=83,tutor-end=93}{\geqslant }\htmlData{tutor-start=93,tutor-end=94}{4}\htmlData{tutor-start=94,tutor-end=95}{,} \end{cases}$ $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{n} \htmlData{tutor-start=10,tutor-end=11}{+} \htmlData{tutor-start=12,tutor-end=13}{5}$ ，第 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个月底的共享单车的保有量是前 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个月的累计投放量与累计损失量的差. (1) 求该地区第 4 个月底的共享单车的保有量； (2) 已知该地共享单车停放点第 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个月底的单车容纳量 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{4}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{4}\htmlData{tutor-start=14,tutor-end=15}{6}\htmlData{tutor-start=15,tutor-end=16}{)}^{\htmlData{tutor-start=18,tutor-end=19}{2}} \htmlData{tutor-start=21,tutor-end=22}{+} \htmlData{tutor-start=23,tutor-end=24}{8}\htmlData{tutor-start=24,tutor-end=25}{8}\htmlData{tutor-start=25,tutor-end=26}{0}\htmlData{tutor-start=26,tutor-end=27}{0}$ (单位：辆). 设在某月底，共享单车保有量达到最大，问该保有量是否超出了此时停放点的单车容纳量？

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 19 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

解答题 · 平面几何

在平面直角坐标系 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中，已知椭圆 $\varGamma: \frac{x^{2}}{4} + y^{2} = 1$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为 $\varGamma$ 的上顶点， $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为 $\varGamma$ 上异于上、下顶点的动点， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴正半轴上的动点. (1) 若 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 在第一象限，且 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=6}{=} \sqrt{\htmlData{tutor-start=13,tutor-end=14}{2}}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的坐标； (2) 设 $\htmlData{tutor-start=0,tutor-end=1}{P}\left(\frac{\htmlData{tutor-start=13,tutor-end=14}{8}}{\htmlData{tutor-start=16,tutor-end=17}{5}}\htmlData{tutor-start=18,tutor-end=19}{,} \frac{\htmlData{tutor-start=26,tutor-end=27}{3}}{\htmlData{tutor-start=29,tutor-end=30}{5}}\right)$ ，若以 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{P}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{M}$ 为顶点的三角形是直角三角形，求 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的横坐标； (3) 若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{M}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{M}\htmlData{tutor-start=9,tutor-end=10}{P}\htmlData{tutor-start=10,tutor-end=11}{|}$ ，直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 与 $\varGamma$ 交于另一点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ ，且 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{Q}} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{2}\overrightarrow{\htmlData{tutor-start=39,tutor-end=40}{A}\htmlData{tutor-start=40,tutor-end=41}{C}}$ ， $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{P}\htmlData{tutor-start=17,tutor-end=18}{Q}} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{4}\overrightarrow{\htmlData{tutor-start=39,tutor-end=40}{P}\htmlData{tutor-start=40,tutor-end=41}{M}}$ ，求直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 的方程.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 20 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

解答题 · 代数

设定义在 $\mathbf{\htmlData{tutor-start=8,tutor-end=9}{R}}$ 上的函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 满足：对于任意的 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}} \htmlData{tutor-start=13,tutor-end=17}{\in }\mathbf{\htmlData{tutor-start=25,tutor-end=26}{R}}$ ，当 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{2}}$ 时，都有 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{)} \htmlData{tutor-start=9,tutor-end=19}{\leqslant }\htmlData{tutor-start=19,tutor-end=20}{f}\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{x}_{\htmlData{tutor-start=24,tutor-end=25}{2}}\htmlData{tutor-start=26,tutor-end=27}{)}$ . (1) 若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{a}\htmlData{tutor-start=8,tutor-end=9}{x}^{\htmlData{tutor-start=11,tutor-end=12}{3}} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{1}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的取值范围； (2) 若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是周期函数，证明： $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是常值函数； (3) 设 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 恒大于零， $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是定义在 $\mathbf{\htmlData{tutor-start=8,tutor-end=9}{R}}$ 上的、恒大于零的周期函数， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 的最大值. 函数 $\htmlData{tutor-start=0,tutor-end=1}{h}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{g}\htmlData{tutor-start=12,tutor-end=13}{(}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{)}$ . 证明：“ $\htmlData{tutor-start=0,tutor-end=1}{h}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是周期函数”的充要条件是“ $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是常值函数”.

题解状态：标准答案与规范题解待补充

题目标签：上海卷第 21 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。